5 UPPER AND LOWER INTEGRALS 115
defined, and is denoted by £/„; also it belongs to £ (12.7.6). If we apply
(13.5.2) to the sequence of partial sums ^fn9 and (13.5.3) to each of the
terms in this partial sum, we obtain the following corollary:
(13.5.4) If(fn)nzi w & sequence of functions ^0 belonging to «/, then
(00 \ 00
!/„ = lA/.).
„=*! / „=!
Now consider an arbitrary mapping / of X into R. There always exist
functions heJ^ such that /z^/, for example the constant function equal
to +00. Put
(13.5.5) M*(/) = inf )u*(fc);
this number £**(/) is called the upper integral of / with respect to the
If /e «/ it is clear that this definition agrees with the preceding one. Here
the value of /**(/) can be any element of R. The relation /^ g implies
/**(/) ^ M*(#)- F°r anY scalar a > 0 we have ti*(af) = <3ju*(/).
(1 3.5.6) If the sumft +/2 o/ /wo mappings ofX into R /51 defined at all points
o/X, and if n*^) > — oo and /J,*(f2) > — oo,
This is obvious if one of the numbers n*(fi)> /**(/2) 1S +°°* If not,
given any a > /^*(/!) and b > ju*(/2), there exist hl9 h2 in */ such that
A ^ /zi, /2 ^ ^2 and ^(/zj) ^ a, ju*(/z2) ^ *• It follows that A4 + A2 ^/i +/2
and /X*(/ZA + /z2) ^ a + £ by (13.5.3). Hence the result.
(13.5.7) If(fn) is any increasing sequence of mappings o/X into E such that
> — oo for all sufficiently large n, then
\ n /
= sup /**(/„) = lim
The inequality ju*/sup/w\ ^ sup u*(fn) is clear. Let us prove the reverse
inequality. We may assume that sup ju*(/M) < +00, otherwise there is
nothing to prove. By hypothesis, we may therefore assume that sup /x*(/n)
and all the ju*(/n) art finite. For each e > 0, we shall show that there exists an> the